Pyramid Problems in Soar & ACT-R. John Laird 26 th Soar Workshop

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1 Pyramid Problems in Soar & ACT-R John Laird 26 th Soar Workshop

2 Big Picture Goals Take instruction (not using NL) Task instructions Problem structure Execute task using domain-independent interpretation No task-specific knowledge in rules Except basic mathematics (7+6 = 13) A few bits of special knowledge for meta-reasoning Match human data and compare to ACT-R Chunking? Meta-Cognition

knowledge in rules Except basic mathematics (7+6 = 13) A few bits of special

3 Mastering an Algebraic Concept Pyramids: There is a notation for writing repeated addition where each term added is one less than the previous: For instance, is written as 5 $ 2 Since = 12 we would evaluate 5$2 as 12 and write 5$2 = 12 The parts of 5 $ 2 are given names: 5 is the base and reflects the number you start with 2 is the height and reflects the number of items you add to the base 5 $ 2 is called a pyramid

would evaluate 5$2 as 12 and write 5$2 = 12 The parts of 5 $ 2 are given names: 5 is the base and reflects

4 Soar: Instructions English (<s1> âction <a10> <a11> <a12> ^next <s2>) (<a10> ^command set ^variable sum ^value 0 ^value-type constant) (<a11> ^command set ^variable term ^value base ^value-type variable) (<a12> ^command set ^variable count ^value 0 ^value-type constant) (<s2> âction <a30> ^next <s3>) (<a30> ^command add ^variable sum ^value term ^value-type variable) (<s3> âction <a6> ^next <s4>) (<a6> ^command goal-test ^relation equal ^variable count ^value height ^value-type variable ^type finished) (<s4> âction <a4> <a5> ^next <s2>) (<a4> ^command decrement ^variable term) (<a5> ^command increment ^variable count) 1. Set sum to 0 Set term to base Set count to 0 2. Add term to sum 3. Test if count = height 4. Decrement Term Decrement Count Goto 2

(<s3> ^action <a6> ^next <s4>) (<a6> ^command goal-test ^relation equal ^variable count ^value height ^value-type variable ^type finished) (<s4> ^action <a4> <a5> ^next <s2>) (<a4> ^command

5 Problem Structure and Example Problem (<ps1> ^name base ^type variable ^next <ps2>) (<ps2> ^name $ ^type symbol ^next <ps3>) (<ps3> ^name height ^type variable ^next <ps4>) (<ps4> ^name = ^type symbol ^next <ps5>) (<ps5> ^name answer ^type variable ^next nil) (<p1> ^value 5 ^type constant ^next <p2>) (<p2> ^value $ ^type symbol ^next <p3>) (<p3> ^value 3 ^type constant ^next <p4>) (<p4> ^value = ^type symbol ^next <p5>) (<p5> ^value? ^type unknown ^next nil)

answer ^type variable ^next nil) (<p1> ^value 5 ^type constant ^next <p2>) (<p2> ^value $ ^type symbol ^next <p3>)

6 Basic Flow Initialize-instruction Initialize-problem Encode [Map problem onto problem structure] Process-symbol, Process-variable, Process-unknown Execute-solve-procedure [Interpret procedure to solve problem] Execute-steps Set, Add, Subtract, Increment, Decrement, Goal-test Next-step Write-answer [Write out the answer] Reflect - [Looks for patterns in problems] Detect first-term - height = last-term Detect balanced problems around 0 Next-problem

Set, Add, Subtract, Increment, Decrement, Goal-test Next-step Write-answer [Write out the answer] Reflect -

7 Evaluation Problems 1. 5 $ = $ = $ = $ = $ = $ = $ $ =

8 Expression Writing Problems $ $ (-1) + (-2) 1$3 12. x + (x 1) + (x 2) + (x 3) + (x 4) x$ (20 1) +. + (20 11) 20$ (15 1) +. + (15 x) 15$x 15. z + (z 1) +. + (z y) z$y

x + (x 1) + (x 2) + (x 3) + (x 4) x$4 13. 20 + (20 1) +.

9 Find the Height Problems $ x = = 15 --> x = $ x = = 55 --> x = $ x =912 x = $ x = = -9 --> x = $ x = = > x =

10 Find the Base Problems 21. x $ 2 =15 guess and check: = 18; = 15 or x + (x - 1) + (x -2) = 15 --> 3x - 3 = 15 --> x = x $ 1 = 15 x = x $ 4 = 35 x = x $ 6 = 35 x = x $ 6 = 0 x = x $ 6 = -7 x = 2

(x - 1) + (x -2) = 15 --> 3x - 3 = 15 --> x = 6 22.

11 Soar Approach to Problem Types Solve: 5$3 = Uses execution procedure Describe: Uses describe procedure (what ACT-R does too) Solve: 6$x=15 Uses execution procedure - stops when answer achieved: Learned stop by doing first set of problems Solve: X$2=15 Impasses on setting Base = X Generate and tests values of X and then solves Must create hypothetical problems If fails, then must generate a new guess Smart generator (based on prior problem, prior guesses)

set of problems Solve: X$2=15 Impasses on setting Base = X Generate and tests values of X and then solves Must

12 Individual Human Data a 3-2a 3-3a 3-4a 3-5a 3-6a 3-1m 3-3m 3-4m 3-5m 3-6m

13 Median Human Data Median

14 Simple Model: Height* Time is proportional to Height Base $ Height =? This is clearly the most important part of the procedure Extend to take into account finding base problem X $ 2 = 15 Simple model of guessing X, modifying guess if wrong.

15 Median, Height* 140 8:1000 $ : 100 $ x = : 10 $ x =55 Median Height* Correlations: H*=.866 removing high values

16 Comments on 1000$2000: John Anderson 1. Students averaged about half of their time in unproductive attempts before they tried a method that work. 2. An unproductive path tried by many was to find an analogy to what they knew about factorial. 3. Five students reasoned about simpler problems like 2$4. 4. Others reasoned more abstractly. 5. A number of students confirmed the answer (0) by a second method before giving it as their final answer. 6. The final ACT-R model tried factorial, then abstract reasoning, and finally confirmed by solving 2$4. 7. Two significant issues for modeling are interrupting regular processing and accumulating needed knowledge. 8. Both are metacognitive in that they require parallel reflection on the ongoing problem solving

A number of students confirmed the answer (0) by a second method before giving it as their final answer. 6.

17 Soar Approach to 1000$2000 Detects large height Attempts abstract solution What can it compute? First-term: 1000, Last-term: (derived from observed relation) Notice balanced : 1000, => 0 Create simple problem to check Creates 2$4= Solve simple problem => 0 Assumes that is the answer Special prior knowledge: Detect large height Note balanced Simple problem generator Soar doesn t mess around with factorial, etc. like ACT model and humans do but clearly could.

problem to check Creates 2$4= Solve simple problem => 0 Assumes that is the answer Special prior knowledge: Detect

18 Median, ACT, Soar (scaled) Median ACT Soar ACT-R=.966 Soar =.907 Ignoring problems 8, 20 ACT/Soar =.97

19 Median, ACT+, Soar, Soar w/ Chunking Median ACT+ Soar chunking

20 140 Chunking with 1PE/Decision ACT-R=.966 Soar =.907 1PE =.906 Ignoring problems 8, Median ACT+ Soar 1-PE C

906 Ignoring problems 8, 20 120 100 80 60 Median

21 Last 6 problems Median ACT-R Soar 1PE Chunk

22 First 7 problems Median ACT+ Soar 1-PE C 5 $ 3 Instructions: 5 $

23 Nuggets: Conclusions Can do instruction taking (again) Leads to surprisingly good results It is (almost) all about doing the task (following instructions) Results hold up with chunking 1PE/Decision Soar is natural for metacognition Coal: Impasses Creating test problems in subgoals Reasoning about structures complex structures (variable attributes) More work to do on detailed comparison with ACT-R More work on where some extra knowledge comes from Soar model is scaled Not 50msec/decision No model of perception,

Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1